Despite recent progress, many works still consider simple geometric domains to solve either forward or inverse problems, hindering the applicability of PINNs to real world problems, where the objects of study may have complicated shapes and topologies. In this area, there have been multiple attempts to introduce complexity to the input domain of the neural networks. One approach is to describe the boundary of the domain with a signed distance function [1]}, [2]}, [3]}. In this way, the boundary conditions can be satisfied exactly instead of relying on a penalty term in the loss function. Another approach is to use domain decomposition to model smaller but simpler domains [4]}. Coordinate mappings between a simple domain have also been proposed for convolutional neural networks [5]}. Nonetheless, all these works demonstrate examples of 2-dimensional shapes. When using PINNs in 3-dimensional surfaces there have been attempts to ensure that the vector fields that may appear in the partial differential equations remain tangent to the surface by introducing additional terms in the loss function [6]}, [7]}. This approach works well when the surface is relatively simple and smooth, specifically when the Euclidean distance of the embedding 3-D space between two points in the domain is similar to the intrinsic geodesic distance on the manifold. However, in several applications the two distances may sensibly differ, as exemplified in Figure REF .
